Rohsenow W., Hartnett J., Young Cho. Handbook of Heat Transfer (776121), страница 5
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The rate equation for mass diffusion is known as Fick's law, and for a transfer ofspecies 1 in a binary mixture it may be expressed asdC1jl --"- D ~(1.6)where C1 is a mass concentration of species 1 in units of mass per unit volume. This expressionis analogous to Fourier's law (Eq. 1.1). Moreover, just as Fourier's law serves to define oneimportant transport property, the thermal conductivity, Fick's law defines a second importanttransport property, namely the binary diffusion coefficient or mass diffusivity D.
The quantityjl [mass/(time x surface area)] is defined as the mass flux of species 1, i.e., the amount ofspecies 1 that is transferred per unit time and per unit area perpendicular to the direction oftransfer. In vector form Fick's law is given asjl = - D V C 1(1.7)In general, the diffusion coefficient D for gases at low pressure is almost composition independent; it increases with temperature and varies inversely with pressure. Diffusion coefficients are markedly concentration dependent and generally increase with temperature.RadiationRadiation, or more correctly thermal radiation, is electromagnetic radiation emitted by abody by virtue of its temperature and at the expense of its internal energy. Thus thermal radiation is of the same nature as visible light, x rays, and radio waves, the difference betweenthem being in their wavelengths and the source of generation.
The eye is sensitive to electromagnetic radiation in the region from 0.39 to 0.78 ~tm; this is identified as the visible region ofthe spectrum. Radio waves have a wavelength of 1 x 10 3 to 2 x 101° ~tm, and x rays have wavelengths of 1 × 10-5 to 2 x 10-2 ktm, while the bulk of thermal radiation occurs in rays fromapproximately 0.1 to l00 ktm. All heated solids and liquids, as well as some gases, emit thermal radiation.
The transfer of energy by conduction requires the presence of a materialmedium, while radiation does not. In fact, radiation transfer occurs most efficiently in a vacuum. On the macroscopic level, the calculation of thermal radiation is based on the StefanB o l t z m a n n law, which relates the energy flux emitted by an ideal radiator (or blackbody) tothe fourth power of the absolute temperature:eb = t~T 4(1.8)Here ~ is the Stefan-Boltzmann constant, with a value of 5.669 × 10-8 W/(m2.K4), or 1.714 x10 -9 Btu/(h.ft 2"°R4). Engineering surfaces in general do not perform as ideal radiators, and forreal surfaces the above law is modified to reade = et~T 4(1.9)The term e is called the emissivity of the surface and has a value between 0 and 1. When twoblackbodies exchange heat by radiation, the net heat exchange is then proportional to the difference in T 4.
If the first body "sees" only body 2, then the net heat exchange from body 1 tobody 2 is given byq = aAI(T~ - T~)(1.10)1.4CHAFFERONEWhen, because of the geometric arrangement, only a fraction of the energy leaving body 1 isintercepted by body 2,q = ~A1F~_2(T 4 - T 4)(1.11)where FI_ 2 (usually called a shape factor or a view factor) is the fraction of energy leavingbody 1 that is intercepted by body 2. If the bodies are not black, then the view factor F~_ 2 mustbe replaced by a new factor ~1- 2 which depends on the emissivity ~ of the surfaces involvedas well as the geometric view. Finally, if the bodies are separated by gases or liquids thatimpede the radiation of heat through them, a formulation of the heat exchange processbecomes more involved (see Chap.
7).ConvectionConvection, sometimes identified as a separate mode of heat transfer, relates to the transferof heat from a bounding surface to a fluid in motion, or to the heat transfer across a flow planewithin the interior of the flowing fluid. If the fluid motion is induced by a pump, a blower, afan, or some similar device, the process is called forced convection. If the fluid motion occursas a result of the density difference produced by the temperature difference, the process iscalled free or natural convection.Detailed inspection of the heat transfer process in these cases reveals that, although thebulk motion of the fluid gives rise to heat transfer, the basic heat transfer mechanism is conduction, i.e., the energy transfer is in the form of heat transfer by conduction within the moving fluid.
More specifically, it is not heat that is being convected but internal energy.However, there are convection processes for which there is, in addition, latent heatexchange. This latent heat exchange is generally associated with a phase change between theliquid and vapor states of the fluid. Two special cases are boiling and condensation.Heat Transfer Coefficient.
In convective processes involving heat transfer from a boundarysurface exposed to a relatively low-velocity fluid stream, it is convenient to introduce a heattransfer coefficient h, defined by Eq. 1.12, which is known as Newton's law ofcooling:q"= h ( T ~ - Tf)Fluid flow(1.12)Here T~ is the surface temperature and Tf is a characteristic fluid temperature.For surfaces in unbounded convection, such as plates, tubes, bodies of revolution, etc.,immersed in a large body of fluid, it is customary to define h in Eq. (1.12) with Tr as the temperature of the fluid far away from the surface, often identified as T~ (Fig.
1.2). For boundedconvection, including such cases as fluids flowing in tubes or channels, across tubes in bundles,etc., Tyis usually taken as the enthalpy-mixed-mean temperature, customarily identified as Tin.The heat transfer coefficient defined by Eq. 1.12 is sensitive to the geometry, to the physical properties of the fluid, and to the fluid velocity.
However, there are some special situationsin which h can depend on the temperature difference AT Tw - TI. For example, if the surface is hot enough to boil a liquid surrounding it, h will typically vary as ATE; or in the caseof natural convection, h varies as some weak power of A T Btypically as AT TM or AT 1/3.It is important to note that Eq. 1.12as a definition of h is valid in these cases too, although itsusefulness may well be reduced.As q " - q/A, from Eq. 1.12 the thermal resistance in convection heat transfer is given byo,I,-o~--T~F I G U R E 1.2 Velocity and temperature distributions in flow over a flat plate.1Rth-hAwhich is actually the resistance at a surface-to-fluid interface.BASICCONCEPTSOF HEATTRANSFER1.5At the wall, the fluid velocity is zero, and the heat transfer takes place by conduction.Therefore, we may apply Fourier's law to the fluid at y = 0 (where y is the axis normal to theflow direction, Fig.
1.2):q " = - k ~9-~YTly=0(1.13)where k is the thermal conductivity of fluid. By combining this equation with Newton's law ofcooling (Eq. 1.12), we then obtainh-q"Tw- T:_k(OT/Oy)ly=0rw- T:_(1.14)so that we need to find the temperature gradient at the wall in order to evaluate the heattransfer coefficient.Similar results may be obtained for convective mass transfer If a fluid of species concentration C1= flows over a surface at which the species concentration is maintained at somevalue Cl.w ~ C1,~, transfer of the species by convection will occur. Species 1 is typically a vaporthat is transferred into a gas stream by evaporation or sublimation at a liquid or solid surface,and we are interested in determining the rate at which this transfer occurs.
As for the case ofheat transfer, such a calculation may be based on the use of a convection coefficient [3, 5]. Inparticular we may relate the mass flux of species 1 to the product of a transfer coefficient anda concentration differenceJ1 = hD(Cl,w-Cl,oo)(1.15)Here hD is the convection mass transfer coefficient and it has a dimension of Lit.At the wall, y = 0, the fluid velocity is zero, and species transfer is due only to diffusion;hencejl--DOC1 I-~yy=0(1.16)Combining Eqs. 1.17 and 1.18, it follows thathD = - D(OC,/Oy)ly=o(1.17)C1, w - Cl,ooTherefore conditions that influence the surface concentration gradient (~Cl/OY)ly=Owill alsoinfluence the convection mass transfer coefficient and the rate of species transfer across thefluid layer near the wall.For convective processes involving high-velocity gas flows (high subsonic or supersonicflows), a more meaningful and useful definition of the heat transfer coefficient is given byq"= h(Tw - Taw)(1.18)Here Taw,commonly called the adiabatic wall temperature or the recovery temperature, is theequilibrium temperature the surface would attain in the absence of any heat transfer to orfrom the surface and in the absence of radiation exchange between the surroundings and thesurface.
In general the adiabatic wall temperature is dependent on the fluid properties andthe properties of the bounding wall. Generally, the adiabatic wall temperature is reported interms of a dimensionless recovery factor r defined asV2Taw = Tf+ r 2Cp(1.19)The value of r for gases normally lies between 0.8 and 1.0. It can be seen that for low-velocityflows the recovery temperature is equal to the free-stream temperature TI. In this case,] .6CHAPTER ONEEq. 1.15 reduces to Eq. 1.12.
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